Quadratic Weighting as an Aggregation-Stable Power Law: Numerical Illustration and Structural Interpretation

The quadratic probability rule plays a central role in quantum theory, yet its structural origin remains conceptually debated. In this work we investigate a simple aggregation principle that naturally selects quadratic weighting. Consider microscopic contributions that combine additively into macroscopic observables, with weights assigned through a generalized power rule M(S)=∣S∣^p. We present a scaling argument showing that within this power-law family the exponent p=2is the only value compatible with extensive scaling when independent contributions aggregate additively in the central-limit regime. To illustrate the structural stability of this exponent, we perform numerical experiments based on block aggregation of complex amplitudes and define an operational coarse-graining map for the effective exponent under aggregation. Across a wide range of initial exponents and amplitude distributions the resulting transformation approaches a fixed point near p≈2. Repeated coarse-graining produces flows toward this value, indicating local attractive stability. A destructive control experiment shows that phase randomization weakens but does not eliminate the quadratic fixed point. These results suggest that quadratic weighting naturally emerges as a stable macroscopic measure for systems in which many microscopic contributions combine additively.